Theorem exlimddOLD 2221

Description: Obsolete version of exlimdd 2220 as of 3-Sep-2023. (Contributed by Mario Carneiro, 9-Feb-2017.) (New usage is discouraged.) (Proof modification is discouraged.)

Hypotheses

Ref	Expression
exlimdd.1	⊢ Ⅎ𝑥𝜑
exlimdd.2	⊢ Ⅎ𝑥𝜒
exlimdd.3	⊢ (𝜑 → ∃𝑥𝜓)
exlimdd.4	⊢ ((𝜑 ∧ 𝜓) → 𝜒)

Assertion

Ref	Expression
exlimddOLD	⊢ (𝜑 → 𝜒)

Proof of Theorem exlimddOLD

Step	Hyp	Ref	Expression
1		exlimdd.3	. 2 ⊢ (𝜑 → ∃𝑥𝜓)
2		exlimdd.1	. . 3 ⊢ Ⅎ𝑥𝜑
3		exlimdd.2	. . 3 ⊢ Ⅎ𝑥𝜒
4		exlimdd.4	. . . 4 ⊢ ((𝜑 ∧ 𝜓) → 𝜒)
5	4	ex 415	. . 3 ⊢ (𝜑 → (𝜓 → 𝜒))
6	2, 3, 5	exlimd 2218	. 2 ⊢ (𝜑 → (∃𝑥𝜓 → 𝜒))
7	1, 6	mpd 15	1 ⊢ (𝜑 → 𝜒)

Syntax hints:   → wi 4   ∧ wa 398  ∃wex 1780  Ⅎwnf 1784

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1970  ax-7 2015  ax-12 2177

This theorem depends on definitions:  df-bi 209  df-an 399  df-ex 1781  df-nf 1785

This theorem is referenced by: (None)